1. Field of the Invention
The present invention pertains to an illumination optical system, exposure apparatus, and microdevice manufacturing method. In particular, the present invention relates to an illumination optical system capable of illuminating a mask, reticle, or other such object; the optical system""s illumination having a uniform illuminance distribution. The present invention further relates to an exposure apparatus equipped with such an illumination optical system and capable of being used, among other things, during the manufacture of semiconductor elements, liquid crystal display elements, image pickup elements, thin-film magnetic heads, and/or other such microdevices. The present invention also relates to a microdevice manufacturing method employing such an exposure apparatus.
2. Background of the Invention
Exposure apparatuses may be employed during photolithographic operationsxe2x80x94such operations representing a portion of the operations for the manufacture of semiconductor elements, liquid crystal display elements, image pickup elements, thin-film magnetic heads, and/or other such microdevicesxe2x80x94to transfer patterns formed on masks or reticles (hereinafter referred to collectively as xe2x80x9cmaskxe2x80x9d) onto wafers, glass plates, substrates or the like (hereinafter referred to collectively as xe2x80x9csubstratexe2x80x9d) which have been coated with photoresist or other such photosensitive material. To be able to illuminate a mask with illumination of uniform illuminance distribution, such an exposure apparatus may be equipped with an illumination optical system capable of causing light emitted from an excimer laser or other such light source to possess a uniform illuminance distribution within a beam formed so as to have a prescribed cross-sectional shape.
If the illuminance distribution of the light emitted from such an illumination optical system varies over the surface of the mask or substrate, there will be nonuniformity in linewidth throughout the pattern formed on the substrate. This variation occurs because the exposure dose of the light irradiating the substrate will vary in correspondence to that illuminance distribution. However, high uniformity in linewidth is demanded during the manufacture of semiconductor elements employed in logic circuits, such semiconductor elements representing one category among the semiconductor elements mentioned above as examples of microdevices. Linewidth uniformity is required because nonuniformity in pattern linewidth will result in decreased operational speed. As an example of the significance of this fact, central processor units (CPUs) operating at frequencies of several GHz have in recent years become standard, and because further increases in operating speed can be expected to be achieved in the future, increased uniformity of pattern linewidth is likely to be extremely important.
To cause light irradiating a substrate to have a uniform exposure over the surface of the substrate, conventional exposure apparatuses have employed illumination optical systems possessing condenser lenses having distortion. The value of the distortion being varied so as to achieve a uniform exposure dose across the surface of the substrate. Referring to FIG. 11, the principle by which the illuminance distribution might be varied by varying distortion of a condenser lens is briefly described. FIG. 11 is a drawing to assist in description of the principle by which an illuminance distribution might be adjusted by means of a condenser lens.
In FIG. 11, P1 represents a light source, 100 represents a condenser lens, and P2 represents the plane of an object to be illuminated (xe2x80x9cobject planexe2x80x9d). This object plane P2 might for example be the plane in which the pattern on a mask is formed. In the discussion below, xcex8 represents the exit angle of a light beam emitted from light source P1 (the exit angle of a light beam emitted so as to be parallel to optical axis AX being taken to have xcex8=0), f represents the focal length of condenser lens 100, and h represents the distance from optical axis AX to a location on object plane P2 at which the light beam emitted from light source P1 at exit angle xcex8 is incident thereon.
Assuming standard Koehler illumination, the relationship describing projection by the condenser lens will, in general, be given by FORMULA (1), below:
h=fxc2x7g(xcex8)xe2x80x83xe2x80x83(1)
Note that at FORMULA (1), above, g(xcex8) is a function of xcex8.
If we assume that light source P1 is a perfectly diffusing surface (a photometrically ideal surface illuminant), then illumination at object plane P2 will be uniform when g(xcex8)=sin(xcex8). We therefore take the distortion of condenser lens 100 to be zero when g(xcex8)=sin(xcex8).
Let us first consider the case in which the distortion of condenser lens 100 is zero. In such a case, the infinitesimal area dS of the locus on object plane P2 of a light beam of infinitesimal solid angle dxcexa9 emitted from light source P1 is given by FORMULA (2), below:
dS=dh d"psgr" h=f2sin xcex8 cos xcex8 dxcex8 d"psgr"xe2x80x83xe2x80x83(2)
. . . where "psgr" is an angle of rotation about optical axis AX.
We next consider the case in which condenser lens 100 has nonzero distortion. The relationship describing projection when there is n% distortion at some image height is given by FORMULA (3), below:
h=fsin xcex8(1xe2x88x92n/100)xe2x80x83xe2x80x83(3)
Now, because the dimensions of condenser lens 100 are fairly uncomplicated, there is little generation of aberration of order three or higher. It is therefore sufficient to likewise only consider distortion attributable to aberration up to the third order. Upon making such an assumption, since distortion is now assumed to be proportional to the square of image height, we can express this in the form n=xcex1sin2xcex8, where xcex1 is a constant.
In such a case, the infinitesimal area dS of the locus on object plane P2 of a light beam of infinitesimal solid angle dxcexa9 emitted from light source P1 is given by FORMULA (4), below:                                                                         d                ⁢                                  xe2x80x83                                ⁢                S                            =                            ⁢                              d                ⁢                                  xe2x80x83                                ⁢                h                ⁢                                  xe2x80x83                                ⁢                d                ⁢                                  xe2x80x83                                ⁢                ψ                ⁢                                  xe2x80x83                                ⁢                h                                                                                                      ⁢                                                f                  2                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                θ                ⁢                                  xe2x80x83                                ⁢                cos                ⁢                                  xe2x80x83                                ⁢                θ                ⁢                                  xe2x80x83                                ⁢                d                ⁢                                  xe2x80x83                                ⁢                θ                ⁢                                  xe2x80x83                                ⁢                d                ⁢                                  xe2x80x83                                ⁢                ψ                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          4                      ⁢                                              xe2x80x83                                            ⁢                      α                      ⁢                                              xe2x80x83                                            ⁢                                              sin                        2                                            ⁢                                              θ                        /                        100                                                              +                                          3                      ⁢                                              xe2x80x83                                            ⁢                                              α                        2                                            ⁢                                              xe2x80x83                                            ⁢                                              sin                        4                                            ⁢                                              θ                        /                        10000                                                                              )                                                                                        (        4        )            
FORMULA (2), above, gives the infinitesimal area dS of the locus on object plane P2 of a light beam of infinitesimal solid angle dxcexa9 emitted from light source P1 for zero distortion at condenser lens 100. FORMULA (4), above, gives the infinitesimal area dS of the locus on object plane P2 of a light beam of infinitesimal solid angle dxcexa9 emitted from light source P1 for nonzero distortion at condenser lens 100.
FORMULAS (2) and (4) determine the infinitesimal areas dS of the loci on object plane P2 produced by light beams of identical infinitesimal solid angle dxcexa9 emitted from light source P1. Using FORMULAS (2) and (4), one obtains a smaller infinitesimal area dS when there is distortion as compared with the infinitesimal area dS obtained when there is no distortion. This is so despite use of the same infinitesimal solid angle dxcexa9. From these results, one can conclude that illuminance will be greater by a corresponding amount.
If we now take the ratio of the expressions at the right sides of FORMULAS (2) and (4), above, we find that infinitesimal area dS is foreshortened due to distortion by a factor given by:
1xe2x88x924xcex1 sin2xcex8/100+3xcex12sin4xcex8/10000.
Since the term 3xcex12sin4xcex8/10000appearing in this formula can be ignored when distortion is exceedingly small, i.e., for xcex1 less than  less than 1, the factor by which infinitesimal area dS is foreshortened due to distortion can in such case be said to be substantially given by:
1xe2x88x924xcex1 sin2xcex8/100xe2x80x83xe2x80x83(5)
The smaller irradiated area for the same identical infinitesimal solid angle exiting the light source means that illuminance will be higher by a corresponding amount. Specifically, from FORMULA (5) we see that when there is distortion at the condenser lens, illuminance will display a distribution proportional to the square of sin xcex8 (xe2x88x9d image height). From FORMULA (5), we can also see that the magnitude of the second-order component of the illuminance distribution produced will be proportional to xcex1, i.e., to the amount of distortion.
The foregoing principle has conventionally been employed to adjust the illuminance distribution by varying the amount of distortion at the condenser lens. Note that whereas we have, in the foregoing description, confined our discussion to the change in the second-order component of the nonuniformity in illuminance produced by a change in the amount of distortion at condenser lens 100, it should of course be understood that the first-order component of the illuminance distribution (the component proportional to image height) may also be adjusted together with the second-order component through use of various other adjustment means.
While the illuminance distribution of the light irradiating the substrate has been adjusted in conventional exposure apparatuses by varying the amount of distortion at a condenser lens as described above, more recently the requirements dictated by uniform linewidth have made it difficult to vary condenser lens distortion. The reason for this is that varying the amount of condenser lens distortion alters the numerical aperture of the illuminating light at every value of image height thereof. This is generally described in Japanese Patent Application Publication Kokai No. H9-22869 (1997).
Now we have already mentioned that in order to form a pattern with uniform linewidth it is necessary that the light irradiating a substrate have uniform illuminance distribution over the surface of the substrate, but it is also necessary that numerical aperture be uniform within the region of exposure at which the substrate is irradiated by light. This is because if numerical aperture is not uniform within the region of exposure, i.e., if numerical aperture varies as a function of location within this exposure region, spatial coherence will be nonuniform, and this will cause the linewidth of the pattern formed on the substrate to be nonuniform.
There is therefore a need in modern exposure apparatuses for a mechanism by which the second-order component of the illuminance distribution might be adjusted without having to vary condenser lens distortion. But because there has not conventionally been a mechanism which would satisfy such requirements, illuminance distribution has conventionally been adjusted in an extremely tedious process in which a plurality of filters having a variety of transmittance distributions are prepared and swapped in so as to minimize the change in condenser lens distortion, with the final fine-tuning being carried out by making minuscule adjustments of condenser lens distortion within a range as permitted by allowed tolerances.
The present invention was conceived in light of the foregoing state of affairs and has as its object the provision of an illumination optical system permitting nonstepwise adjustment of the second-order component of the illuminance distribution without the need for any alteration in condenser lens distortion whatsoever, an exposure apparatus equipped with such an illumination optical system, and a microdevice manufacturing method in which microdevice manufacture is carried out by employing such an exposure apparatus to form a highly detailed pattern.
In order to solve one or more of the foregoing problems, the present invention sets forth an illumination optical system for illuminating one or more objects with a light emitted from one or more light sources. The illumination optical system has two or more filter members located in one or more optical paths of the light emitted from at least one of the light. The optical system further has transmittance distributions substantially definable by one or more functions of order three or higher with respect to transmittance as a function of position in one or more directions substantially transverse to at least one or more of the optical paths.
In accordance with this aspect of the present invention, because two or more filter members arranged in optical paths have transmittance distributions substantially definable by functions comprising functions of order three or higher with respect to transmittance as a function of position in directions substantially transverse to optical paths, it is possible to adjust second-order components of illuminance distributions through use of filters alone. Furthermore, the combination of such transmittance distributions permits attainment of combined transmittance distributions which can be varied after the fashion of second-order functions with respect to transmittance as a function of position in directions substantially transverse to optical paths. And because it is thus possible to adjust second-order components of illuminance distributions using filters alone, there is no longer any need whatsoever for the sort of varying of condenser lens distortion which was necessary for adjustment of the second-order component of the illuminance distribution. Furthermore, it is possible to eliminate the change in numerical aperture of illuminating light at every value of image height thereof which occurs as a result of varying of condenser lens distortion without losing the ability to adjust the second-order component of the illuminance distribution. This is extremely favorable from the standpoint of ability to generate a pattern having uniform linewidth.
Furthermore, in the illumination optical system in accordance with the first aspect of the present invention, it is preferred that at least one of the transmittance distributions substantially definable by one or more functions comprising one or more functions of order three or higher be a transmittance distribution substantially definable by one or more functions comprising a third-order function. Moreover, the illumination optical system may further include one or more drive apparatuses capable of moving at least one of the two or more filter members in one or more directions substantially transverse to at least one of the one or more optical paths.
This embodiment of the illumination optical system is further characterized in that at least one of the one or more drive apparatuses may be capable of moving at least one of the two or more filter members continuously in one or more directions substantially transverse to at least one of the one or more optical paths. In accordance with this aspect of the present invention, because filter members may be moved continuously in directions substantially transverse to optical paths, it is possible to adjust illuminance distribution quickly and with high precision.
Moreover, this embodiment of the invention is further characterized in that it may further comprise one or more control apparatuses capable of causing at least one of the one or more drive apparatuses to be driven in such fashion as to permit control of the positional relationship between at least two of the two or more filter members in one or more directions substantially transverse to at least one of the one or more optical paths.
In order to solve one or more of the foregoing problems, an illumination optical system in accordance with a second aspect of the present invention is characterized in that, it includes two or more filter members located in one or more optical paths of light emitted from at least one or more light source. The illumination optical system further has transmittance distributions substantially definable by one or more functions having one or more functions of order three or higher with respect to transmittance as a function of position in respectively at least a first direction which is substantially transverse to at least one of the one or more optical paths and a second direction Y which is substantially perpendicular to the first direction.
In accordance with this aspect of the present invention, because two or more filter members arranged in optical paths have transmittance distributions substantially definable by functions comprising functions of order three or higher respectively with respect to transmittance as a function of position in first directions substantially transverse to optical paths and second directions, it is possible to adjust second-order components of illuminance distributions through use of filters alone. Furthermore, the combination of such transmittance distributions permits the attainment of combined transmittance distributions which can be varied after the fashion of second-order functions with respect to transmittance as a function of position in directions substantially transverse to optical paths. In addition, because it is thus possible to adjust second-order components of illuminance distributions using filters alone, there is no longer any need whatsoever for the sort of varying of condenser lens distortion which was necessary conventionally for adjustment of the second-order component of the illuminance distribution. Furthermore, it is consequently possible to eliminate the alteration in numerical aperture of illuminating light at every value of image height thereof which occurs as a result of varying of condenser lens distortion without losing the ability to adjust the second-order component of the illuminance distribution. This is extremely favorable from the standpoint of ability to generate a pattern having uniform linewidth. And because it is possible to respectively adjust such second-order components in first and second directions, accommodation of a wide variety of illuminance distributions is permitted.
Furthermore, in the illumination optical system in accordance with the second aspect of the present invention, it is preferred that at least one of the transmittance distributions substantially definable by one or more functions comprising one or more functions of order three or higher be a transmittance distribution substantially definable by one or more functions comprising a third-order function. Moreover, this illumination optical system further comprises one or more drive apparatuses capable of moving at least one of the two or more filter members in at least one direction substantially identical with or substantially parallel to at least one of the first or second directions.
An illumination optical system in accordance with the second aspect of the present invention is furthermore characterized in that at least one of the one or more drive apparatuses may be capable of moving at least one of the two or more filter members continuously in at least one direction substantially identical with or substantially parallel to at least one of the first or second directions. In accordance with this aspect of the present invention, because filter members may be moved continuously in directions substantially identical with or substantially parallel to the first or second directions, it is possible to adjust illuminance distribution quickly and with high precision in these directions.
An illumination optical system in accordance with the second aspect of the present invention is in addition characterized in that it may further comprise one or more control apparatuses capable of causing at least one of the one or more drive apparatuses to be driven in such fashion as to permit control of the positional relationship between at least two of the two or more filter members in at least one direction substantially identical with or substantially parallel to at least one of the first or second directions. It is preferred that at least one of the two or more filter members be capable of being arranged near at least one of the one or more objects to be illuminated and/or substantially in or near a plane optically conjugate to a plane more or less containing at least one of the one or more objects to be illuminated.
It is moreover favorable in the illumination optical systems in accordance with the first and second aspects of the present invention that at least two of the two or more filter members be capable of being arranged so as to have respective transmittance distributions in more or less mutually inverse relationship with respect to transmittance as a function of position in one or more directions substantially transverse to at least one of the one or more optical paths.
In order to solve one or more of the foregoing problems, an exposure apparatus in accordance with a first aspect of the present invention is characterized in that, in the context of an exposure apparatus for illuminating one or more masks with a light from one or more light sources and transferring one or more patterns formed on at least one of the one or more masks to one or more photosensitive substrates W, such exposure apparatus comprises one or more mask stages constructed so as to permit at least one of the one or more masks to be loaded thereon. The exposure apparatus further has one or more substrate stages constructed so as to permit at least one of the one or more photosensitive substrates to be loaded thereon. The apparatus further includes one or more illumination optical systems in accordance with the first or second aspects of the present invention and capable of illuminating at least one of the one or more masks with light from at least one of the one or more light sources.
In order to solve one or more of the foregoing problems, an exposure apparatus in accordance with a second aspect of the present invention includes one or more mask stages constructed so as to be capable of movement while at least one of the one or more masks is loaded thereon. The exposure apparatus further includes one or more substrate stages constructed so as to be capable of movement while at least one of the one or more photosensitive substrates is loaded thereon. There is in addition, one or more illumination optical systems according to the first aspect of the present invention and capable of illuminating at least one of the one or more masks with a light from at least one of the one or more light sources. In addition the apparatus had one or more projection optical systems capable of forming on at least one of the one or more photosensitive substrates at least one image of at least one of the one or more patterns on at least one of the one or more masks R.
The exposure apparatus in accordance with the second aspect of the present invention further contains one or more mask stage drive systems coupled to at least one of the one or more mask stages and capable of causing at least one of the one or more mask stages to move. There is also included one or more substrate stage drive systems coupled to at least one of the one or more substrate stages and capable of causing at least one of the one or more substrate stages to move. The apparatus in this embodiment further has one or more controllers coupled to at least one of the one or more mask stage drive systems and at least one of the one or more substrate stage drive systems 41. The controllers are capable of controlling at least one of the one or more mask stage drive systems and at least one of the one or more substrate stage drive systems such that at least one of the one or more masks and at least one of the one or more photosensitive substrates are made to move in one or more directions substantially identical with or substantially parallel to one or more scan directions in correspondence to at least one magnification of at least one of the one or more projection optical systems PL. At least one of the one or more directions is substantially transverse to at least one of the one or more optical paths being capable of being set so as to be substantially transverse to one or more directions corresponding to at least one of the one or more scan directions.
As used herein, xe2x80x9cdirections corresponding to scan directionsxe2x80x9d refers to directions substantially identical with or substantially parallel to projections of scan directions onto filter members by portions of optical systems between masks and filter members, inclusive.
In order to solve one or more of the foregoing problems, a microdevice manufacturing method in accordance with the first aspect of the present invention is characterized in that it comprises an expose step wherein at least one exposure apparatus in accordance with the first and/or second aspects of the present invention is used to expose at least one of the one or more photosensitive substrates W so as to form thereon one or more complete and/or partial latent images of at least one of the one or more patterns present on at least one of the one or more masks. The method also includes a developing step wherein at least one of the one or more latent images on at least one of the one or more photosensitive substrates W is developed.
Some of the various principles behind operation of the present invention will now be described. Here, for convenience of description, we take the case of an illumination optical system comprising two filter members having transmittance distributions representable by third-order power series with respect to transmittance as a function of position in respectively a first direction (x direction) transverse to an optical path and a second direction (y direction) perpendicular to the first direction. As used herein, a xe2x80x9cthird-order power seriesxe2x80x9d is in general a function of the form T=ax3+bx2+cx+d, where a, b, c, and d are constants.
If we assume that the transmittance distributions of the filter members are functions of x and y, then a filter member having a transmittance distribution T(x, y) representable by a third-order power series will be described by FORMULA (6):
T(x, y)=ax3+bx2+cx+ey3+fy2+gy+dxe2x80x83xe2x80x83(6)
. . . where a, b, c, d, e, f, and g are constants.
To further simplify our description, in the description that follows we consider the case of a filter member having a transmittance distribution T(x, y) as described by FORMULA (7):
T(x, y)=ax3+dxe2x80x83xe2x80x83(7)
Two filter members having transmittance distributions T(x, y) as described by FORMULA (7), above, might be prepared, and one might be arranged such that it is rotated 180xc2x0 with respect to the other in the xy plane. Such an arrangement will result in one of the two filter members having a transmittance distribution which we can write as T(x, y)=xe2x88x92ax3+d. The combined transmittance distribution T1 of the two filter members will, in such a case, be given by FORMULA (8):
T1=(ax3+d)(xe2x88x92ax3+d)==xe2x88x92a2x6+d2xe2x80x83xe2x80x83(8)
To simplify the mathematical analysis that follows and to clarify our description, we now introduce suitable approximations for transmittance. Consider the relationship indicated by FORMULA (9), below, when the values of xcex1 and xcex2 are very close to 1.
(xcex1xe2x88x921)(xcex2xe2x88x921)=xcex1xcex2xe2x88x92xcex1xe2x88x92xcex2+1xe2x80x83xe2x80x83(9)
We can arrange FORMULA (9) to get FORMULA (10):
xe2x80x83xcex1xcex2xe2x88x92(xcex1xe2x88x921)(xcex2xe2x88x921)=xcex1+xcex2xe2x88x921xe2x80x83xe2x80x83(10)
But because we have assumed (for purposes of description) that the values of xcex1 and xcex2 are very close to 1, it is clear in such case that (xcex1xe2x88x921)(xcex2xe2x88x921) will be much smaller than xcex1xcex2. FORMULA (10) can, in such case, therefore be written in the form indicated by FORMULA (11):
xcex1xcex2≅xcex1+xcex2xe2x88x921xe2x80x83xe2x80x83(11)
If we assume that the transmittances of the two filter members are very close to 1 (i.e., that transmittance is on the order of 95% to 100%), we can use the relationship at FORMULA (11) to rewrite FORMULA (8), above, as indicated at FORMULA (12), below, which, while being an approximation is nonetheless extremely simple in form.
T1=(ax3+d)(xe2x88x92ax3+d)≅(ax3+d)+(xe2x88x92ax3+d)xe2x88x921=2dxe2x88x921xe2x80x83xe2x80x83(12)
Next, if one of the two filter members is displaced by an amount j in the xe2x88x92x direction and the other is displaced by an amount j in the +x direction so as to obtain an arrangement wherein the two filter members are parallel to the optical axis, but occupy locations displaced by some small amount (here, 2j) with respect to each other, the combined transmittance distribution T2 will in this case be given by FORMULA (13), below.                                                         T2              =                            ⁢                                                {                                                                                    a                        ⁡                                                  (                                                      x                            +                            j                                                    )                                                                    3                                        +                    d                                    }                                ⁢                                  {                                                            -                                                                        a                          ⁡                                                      (                                                          x                              -                              j                                                        )                                                                          3                                                              +                    d                                    }                                                                                                        ≅                            ⁢                                                {                                                                                    a                        ⁡                                                  (                                                      x                            +                            j                                                    )                                                                    3                                        +                    d                                    }                                +                                  {                                                            -                                                                        a                          ⁡                                                      (                                                          x                              -                              j                                                        )                                                                          3                                                              +                    d                                    }                                -                1                                                                                        =                            ⁢                                                6                  ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  j                  ⁢                                      xe2x80x83                                    ⁢                                      x                    2                                                  +                                  2                  ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                                      j                    3                                                  +                                  2                  ⁢                  d                                -                1                                                                        (        13        )            
Subtracting FORMULA (12) from FORMULA (13) allows us to determine the change in transmittance produced as a result of causing the two filter members to be displaced relative to one another in the x direction, which we write as FORMULA (14):
T2xe2x88x92T1=6ajx2+2aj3xe2x80x83xe2x80x83(14)
Displacing two filter members relative to one another thus causes a change in the second-order transmittance distribution. Furthermore, as can be seen from FORMULA (14), the amount of the change in the second-order transmittance distribution is proportional to the amount j by which the filter members are displaced. Note that while FORMULA (14), above, includes a constant component equal to 2aj3, meaning that the transmittance is itself offset by an amount which varies as a function of j, because the value of the constant xe2x80x9caxe2x80x9d is in reality sufficiently small relative to the value of the constant xe2x80x9cd,xe2x80x9d the amount of this offset will not present a problem in practice.
As described above, by arranging two filter members having transmittance distributions representable by third-order power series such that they are substantially coincident with or substantially parallel to an optical path, and by displacing one such filter relative to the other in a direction substantially transverse to such optical path, it is possible to achieve a composite filter which is capable of correcting second-order components of transmittance distributions. It is therefore possible to correct second-order components of illuminance distributions, without having to vary the amount of condenser lens distortion, by arranging these two filter members near the object plane or substantially in or near a plane optically conjugate to the object plane.
Note that as mentioned above and for convenience of description, the foregoing discussion treats the case in which the filter members respectively have transmittance distributions of the form T(x, y)=ax3+d. More generally, a filter member capable of being used in the present invention may have transmittance distribution of the form:
T(x, y)=ax3+bx2+cx+d
. . . where a, b, c, and d are constants.
Moreover, a filter member capable of being used in the present invention may have transmittance distribution as indicated at FORMULA (6), above; i.e.,
T(x, y)=ax3+bx2+cx+ey3+fy2+gy+d
. . . where a, b, c, d, e, f, and g are constants.
In such a case, second-order components of illuminance distributions may be independently adjusted in the two directions x and y by displacing respective filter members relative to one another in the two directions x and y. Furthermore, whereas the foregoing description treats the example of the case where filter members have transmittance distributions representable by third-order power series, components representable by higher-order power series and/or components representable for example by trigonometric functions and/or other such functions may alternatively or additionally be present.
Furthermore, there being no reason that the number of filter members must be limited to two as in the foregoing description, a similar effect may be achieved through combinations of various pluralities of filter members. For example, independent adjustment of second-order components of illuminance distributions in the x direction and second-order components of illuminance distributions in the y direction may be easily achieved through combination of four filters respectively having the transmittance distributions Ta(x, y), Tb(x, y), Tc(x, y), and Td(x, y), below.
Ta(x, y)=ax3+d
Tb(x, y)=xe2x88x92ax3+d
Tc(x, y)=ay3+d
Td(x, y)=xe2x88x92ay3+d
Moreover, whereas the foregoing description confined itself to adjustment of second-order components of illuminance distributions, because transmittance distributions of the filter members of the present invention may have substantial second-order components, it is possible, through collective decentration of such filter combinations taken as a whole, to correct first-order components of illuminance distributions.
Furthermore, the filter members having such transmittance distributions may be prepared using optical thin films or the like, with film design parameters being varied as a function of location. Alternatively, the filter members may be prepared by vapor deposition of light-occluding or light-attenuating microdots of size on the order of or smaller than the limit of resolution on appropriate stock, with the density of such microdots being varied as a function of location; and so forth. There is in fact no particular limitation with regard to the method by which the filter members having such transmittance distributions are prepared. In the event that filters having such transmittance distributions are prepared by varying the probability of existence of microdots as a function of location, it is desirable that there be no particular order to the arrayal of microdots (i.e., that locations having identical transmittances on the respective filter members not have identical arrayal of dots); or where there is a particular order, it is desirable that such particular order be different from filter member to filter member.
Additional objects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. The objects and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the appended claims.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention, as claimed.
The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate several embodiments of the invention and together with the description, serve to explain the principles of the invention.